The Sacrifice of the Mass is the pure offering spoken of by the Prophet Malachi: “For from the rising of the sun to its setting my name is great among the nations, and in every place … is offered to my name … a pure offering” (Malachi 1:11).
It is the nature of worship to make sacrificial offerings to God in thanksgiving and for sins. Jesus made the sacrificial offering once for all time to God for our sins by his passion and death on the cross. The sacrifice of the Mass is our sacrificial offering to God in thanksgiving (=Eucharist).
The question then becomes: “What shall I render to the LORD for all his bounty to me? I will lift up the cup of salvation” (Psalm 116:12-13). The purest and most perfect offering we could possibly make to God the Father is his Son, our Lord Jesus Christ: his now glorified body, his blood that was poured out for us, his soul and his divinity (Hebrews 12:24). And this is possible because Jesus gave the Apostles the power and authority to “do this” at the Last Supper whenever they and their ordained successors (our bishops and priests) repeat Jesus’ words of consecration: “This is my body…my blood” over bread and wine.
When we gather at Mass, we bring forward our imperfect gifts of bread and wine and through the power of the Holy Spirit and the words of Jesus spoken by the priest, they become (are transubstantiatied into) our risen Lord Jesus Christ himself but still retaining their former appearance as bread and wine as at the Last Supper. (There have been Eucharistic miracles where the bread and wine have actually transformed into flesh and blood but that’s another story). This most pure and perfect gift then is offered to God the Father in thanksgiving. After that, as with many Old Testament sacrificial offerings, the holy offering is then eaten by the priest and by us for whom the thanksgiving sacrifice is offered. In this way, we also fulfill Jesus’ command to eat his flesh and drink his blood found in John 6.
If you read 1 Cor 10:16-22 carefully, you should be able to see Paul contrasting three different sacrifices, that of the Jews, of pagans, and of the Christians:
Of Jews, “partaking of the altar [of God]” by eating Israel’s “sacrifices” [offered to God] (v. 18);
Of pagans, “partaking of the table of demons” by eating pagan’s “sacrifices offered to idols…demons” (v. 19-21);
Of Christians, “partaking of the table of the Lord” by eating the Christian sacrifice of broken “bread” and blessed “wine,” i.e., “the body of Christ” and “the blood of Christ,” offered “to God” (v. 16-17, 21)
This is at least my understanding. I hope it is correct and helpful.
Todd